Inverse Problem of Electrodynamics for Anisotropic Medium: Linear Approximation

Abstract

For electrodynamic equations with permittivity specified by a symmetric matrix $$\varepsilon (x) = ({{\varepsilon }_{{ij}}}(x),i,j = 1,2,3)$$ , the inverse problem of determining this matrix from information on solutions of these equations is considered. It is assumed that the permittivity is a given positive constant $${{\varepsilon }_{0}} > 0$$ outside a bounded domain $$\Omega \subset {{\mathbb{R}}^{3}}$$ , while, inside $$\Omega $$ , it is an anisotropic quantity such that the differences $${{\varepsilon }_{{ij}}}(x) - {{\varepsilon }_{0}}{{\delta }_{{ij}}} = :{{\tilde {\varepsilon }}_{{ij}}}(x),$$ $$i,j = 1,2,3,$$ are small. Here, $${{\delta }_{{ij}}}$$ is the Kronecker delta. The inverse problem is studied in the linear approximation. The structure of the solution to a linearized direct problem for the electrodynamic equations is investigated, and it is proved that all elements of the matrix $$\tilde {\varepsilon }(x) = {{\tilde {\varepsilon }}_{{ij}}}(x),\;i,j = 1,2,3$$ , can be uniquely determined by special observation data. Moreover, the problem of recovering the diagonal components $${{\tilde {\varepsilon }}_{{ij}}}(x),\;i = 1,2,3,$$ leads to a usual X-ray tomography problem, so these components can be efficiently computed. The recovery of the other components leads to a more complicated algorithmic procedure.